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Notes on Propositional Logic and First Order Logic

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Notes on Propositional Logic and First Order Logic Notes on propositional logic and first order logic M. Viale Contents 1 Propositional logic3 1.1 Semantics.....................................3 1.2 Disjunctive normal forms............................8 1.3 Proof systems...................................9 1.4 The sequent calculs LK ..............................9 1.5 Exercises on propositional logic and LK-calculus................ 22 2 Basics of first order logic 23 2.1 Examples of first order languages........................ 24 2.2 Syntax and semantics for arbitrary first order languages........... 31 2.3 Free and bounded variables and substitution of symbols inside formulae: what are the problems to match?........................ 36 2.4 Syntactic complexity of terms and formulae.................. 39 2.5 Free and bounded variables of a formula.................... 40 2.6 Basic rules for logic equivalence and prenex normal forms of formulae... 45 2.7 Substitution of terms inside formulae...................... 48 2.8 Geometric interpretation of formulae...................... 49 2.9 Definable sets with and without parameters.................. 52 2.10 Exercises on first order logic........................... 54 3 More on first order logic 55 3.1 First order LK-calculus.............................. 55 3.2 Satisfiable theories and compactness...................... 57 3.3 Classes of L-structures.............................. 59 3.4 Substructures, morphisms, and products.................... 60 3.5 Elementary equivalence and completeness................... 68 3.6 More exercises on first order logic........................ 69 4 Compactness 69 4.1 Proof of the compactness theorem....................... 69 5 First order logic and set theory 79 1 WARNING: These notes are a compendium to Alessandro Andretta's textbook [1] meant as a complement to [1, Chapter I.3]. In particular sections2 and3 of these notes consists of the material covered in [1, Chapter I.3] and our presentation draws heavily from it. We invite the reader to look at [1, Chapter I.3] as a further source of reference for our treatment of the basics of first order logic. 2 1 Propositional logic Propositional logic formalizes in a mathematically rigorous theory certain concepts and procedures which rule our way to reason about mathematics. For example by means of propositional logic we can give a mathematically precise counterpart of the concepts of theorem, mathematical truth, contradiction, logical deduction, equivalence of meaning between two different linguistic expressions, etc.... Nonethe- less propositional logic is still a theory too weak to develop a mathematical theory which reflects all kind of mathematical reasoning. This can be accomplished to a really satisfactory extent by means of first order logic, whose basic properties will be introduced in the second part of these notes. We believe that a short introduction to propositional logic can help to understand the key ideas which leads to develop a rigorous and effective mathematical theory of mathematical reasoning. Definition 1.1. Given an infinite list of propositional variables PropVar = fAi : i 2 Ng, a propositional formula is defined by induction using the following clauses out of propositional variables and connectives f:; ^; _; !g: • each propositional variable Ai is a formula, • if φ is a formula (:φ) is a formula, • if φ, are formulae, also (φ ^ ); (φ _ ); (φ ! ) are formulae. We let Form be the set of propositional formulae. Given a formula φ the set of its propositional variables propvar(φ) is given by the propositional letters occurring in φ. 1.1 Semantics We assign truth values to propositional formulae according to the following defini- tion: Definition 1.2. Let v : PropVar ! f0; 1g. We extend v to a unique map v : Form ! f0; 1g satisfying: • v((:φ)) = 1 − v(φ), • v((φ ^ )) = v(φ) · v( ), • v((φ _ )) = max fv(φ); v( )g, • v((φ ! )) = v(:φ _ ). The intended meaning of the above definition being: we regard 1 as a truth and 0 as a falsity. We are interested to study only propositions which have a definite truth value, such as \There are infinitely many prime numbers", \Every continuous functions f : R ! R is bounded", \Every continuous functions f : [0; 1] ! R is bounded",. , we know that the first and the third statements are true, while the second is false (as witnessed for example by f(x) = x2). Propositional logic is not suited to study propositions for which we are not able to assign a definite truth value among true and false; an example of a statement on which propositional logic is not able to say much is the following: \This sentence 3 is false". The latter statement can be neither true (otherwise it asserts its falsity), nor false (otherwise it would be true). A basic intuition regarding valuations is that a propositional variable A can range among all propositions which are either true or false and that a valuation v decides whether we assign to A a true proposition or a false one. Under the decision made by v regarding the propositional variables, the connectives give a truth value to the other formulae reflecting certain propositional constructions typical of the natural language: • (:φ) stands for the negation of the formula/proposition φ, • (φ ^ ) stands for the conjuction of the formulae/propositions φ, , • (φ _ ) stands for the disjuction of the formulae/propositions φ, , • (φ ! ) stands for the statement \Whenever φ holds also holds". Remark 1.3. A key observation is that the truth value assigned to a formula φ by a valution v depends only on the truth value the valuation assigns to the free variable of φ, i.e.: v0 varprop(φ) = v1 varprop(φ) entails that v0(φ) = v1(φ). This allows to define finite truth tables for each formula φ with 2jvarprop(φ)j rows1 (one row for each possible assignment of 0; 1 to the propositional variables of φ), and jvarprop(φ)j + 1 columns (one column for φ and another column for each of the propositional variables of φ). For example let φ be the formula ((B ! A) ^ ((B _ C) ! A)) with propositional variables A; B; C. Its truth table can be computed as follows: ABC (B ! A) (B _ C) ((B _ C) ! A) φ 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 1 We may omit the truth values of the columns for the subformulae of φ, which are (B ! A); (B _ C); ((B _ C) ! A) and obtain the smaller table in 4 columns (3 for the propositional variables A; B; C occurring in φ and 1 for φ), and 8 rows (as many as the 23 possible disinct as- signments of truth values to the three propositional variables A; B; C occurring in φ): 1jXj denotes the number of elements of the set X 4 ABC φ 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 Remark however that the truth table of all subformulae of φ needs to be computed in order to be able to compute the one of φ. Remark 1.4. A formula φ is built in a finite number of stages starting from the propositional variables and introducing at each stage some propositional connec- tive attached to one or two simpler sub-formulae of φ; parentheses are needed to understand in which order the process of building φ occurs. φ ≡ ((A ^ B) ^ C) is a formula built in two steps out of the three variables A; B; C: first we build ≡ (A ^ B) out of the propositional variables A; B and then φ ≡ ( ^ C) out of and C. Due to the fact that we are interested in formulae up to logical equivalence (i.e. we identify formulae having the same truth table, see Def. 1.11), and we do not want to burden our notation, we will drop parentheses when we believe this cannot cause confusion in the intended meaning of the formula. For example A ^ B ^ C can stand either for ((A ^ B) ^ C) or for (A ^ (B ^ C)), since the two formulae are logically equivalent. On the other hand a writing of the form A^B_C is ambiguous, since it could stand either for the formula θ0 ≡ ((A ^ B) _ C) or for the formula θ1 ≡ (A ^ (B _ C)), which are not logically equivalent. In such cases we will keep enough parentheses to avoid ambiguities, i.e. for θ0 we will write (A^B)_C and for θ1 we will write A ^ (B _ C) dropping in both cases the most external parentheses but keeping the ones that clarifies the priorities on the order of introduction of the connectives. So from now on we will write :φ rather than (:φ), φ ^ , φ _ , φ ! rather than (φ^ ), (φ_ ), (φ ! ), φ1 ^:::^φn rather than (::: ((φ1 ^φ2)^:::)^φn) (or any of the possible rearrangements of the parentheses yielding a formula logically equivalent to the conjunction of all the φi). Definition 1.5. A propositional formula φ is: • a tautology if v(φ) = 1 for all valuations v : VarProp ! f0; 1g, • a contradiction if v(φ) = 0 for all valuations v : VarProp ! f0; 1g, • satisfiable if v(φ) = 1 for some valuation v : VarProp ! f0; 1g. Remark 1.6. φ is a tautology if and only if :φ is a contradiction, φ is satisfiable if and only if it is not a contradiction. A tautology is a proposition which is true regardless of the context in which it is meaningfully interpreted, a contradiction is a proposition which is false regardless of the context in which it is meaningfully interpreted, a satisfiable proposition is a proposition which is true in some of the contexts in which it can be meaningfully interpreted.
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